Semiclassical Inversion Research Articles

Spectroscopic signatures of bond-breaking internal rotation. II. Rotation-vibration level structure and quantum monodromy in HCP

The rotation-vibration level structure of ground electronic state HCP is investigated at vibrational energies approaching and exceeding that of the linear CPH saddle point. With respect to energies above the saddle point, we investigate possible spectroscopic manifestations of strong Coriolis interactions between the hindered, bond-breaking internal rotation of the hydrogen about the CP core and the rotation of the molecule in the space-fixed axis system. With respect to energies below the saddle point, we provide new interpretations, from quantum and semiclassical points of view, of previously observed anomalously large B (rotational) and g22 (energy dependence on the vibrational angular momentum) constants for the large-amplitude pure bending states of HCP (referred to elsewhere as “isomerization” or saddle node states). We also predict similar anomalies in other spectroscopic constants, including the “centrifugal distortion” constant D and the “rotational l-resonance” parameter q2. These changes in the effective spectroscopic rotation-vibration constants are shown to be a direct consequence of the spherical pendulum topology of the HCP bend/internal rotor system, which is associated with a phenomenon called quantum monodromy, defined as the absence of a smoothly valid set of quantum numbers for all states. Our semiempirical model for the HCP bend/internal rotor mode is derived using principles of semiclassical inversion and provides new insights into the breakdown in the ability of rovibrational effective Hamiltonians to model highly vibrationally excited states of HCP.

Algebraic and coordinate space potentials from heavy ion scattering.

An inversion scheme is presented to derive the potentials of algebraic scattering theory from the corresponding S functions. Representative heavy ion scattering data of $^{12}\mathrm{C}$, $^{14}\mathrm{N}$, and $^{16}\mathrm{O}$ ions on $^{208}\mathrm{Pb}$, accurately fitted by McIntyre strong absorption type S functions, are employed to obtain exact algebraic potentials and to generalize the analytical shapes proposed previously by Alhassid et al. The coordinate space potentials corresponding to a number of S functions are also obtained via semiclassical inversion.

Semiclassical and quantal inversion of nuclear scattering at fixed energy

The semiclassical and quantal inversion problems at fixed energy are solved for a number of nuclear scattering systems at various energies. The scattering function is represented as a particular rational or nonrational function of angular momentum. At low energies the semiclassical inversion generally breaks down; it works, however, surprisingly well in many cases.

Potential energy curves for the A 1Σ+u and C 1Σ+g states of He2 obtained by combining scattering, spectroscopy, and a b i n i t i o theory

Crossed beam scattering experiments on He*(21S)+He at energies of 0.95, 1.50, and 2.23 kcal/mol have been combined with spectroscopic and ab initio theoretical data to yield reasonably unique potential energy curves for the A 1Σ+u and C 1Σ+g excited states of He2 over a range of interatomic distances from 0.7 to 7 Å. In this energy range the scattering is particularly sensitive to the long-range barrier in the A state; the resulting barrier height is 1.158±0.02 kcal/mol and the position of the maximum 3.132±0.02 Å. At longer range the potentials are determined by a semiclassical inversion method described earlier [J. Chem. Phys. 69, 4634 (1978)] and the classical theory that underlies this method is used to interpret the complicated scattering patterns at higher energy.

Improvement of the RKR procedure obtained by semi-classical calculations

In this letter we present a new technique to improve RKR molecular potential curve calculations. The potential curve obtained gives calculated G(ν) and B(ν) values extremely close to the measured ones. The aim of these calculations is the same as that of the inverted perturbation approach of Vidal and Sheingraber, but the potential-well corrections are calculated by using Watson's semi-classical inversion procedure.

Comment on “the helium interatomic potential and the undulatory velocity dependence of the low-energy 4he– 4he cross section” by P.G. Burton

The recent claim of Burton that the well-known semiclassical inversion of 4He 2 backward glory undulations gives dubious results is shown to be incorrect.

Unique determination of the He2 ground state potential from experiment by use of a reliable potential model

Measured backward glory oscillations of integral 4He2 and 3He2 scattering cross sections are evaluated by use of an improved semiclassical backward glory formula yielding the energy dependence of the s phases, which allows the calculation of the He2 potential in the region 1.83–2.12 Å via Miller’s semiclassical inversion method. A physically realistic two-parameter potential model, which uses all ab initio data available with sufficient accuracy, is fitted to the inversion result, giving in a unique way the entire helium pair potential function. A well with a depth of 10.74 K at 2.975 Å is obtained, which supports a bound state for 4He2 very near to the dissociation limit. Via construction, the potential model also gives results for individual interaction energy terms in the symmetry adapted perturbation scheme for the He2 interaction. Calculations with the determined potential reproduce the various experimental data available for helium. The best ab initio He2 potentials available today converge towards that potential.

ChemInform Abstract: SEMICLASSICAL INVERSION OF TORSIONAL VIBRATION ENERGY LEVELS: METHOD, AND APPLICATION TO ACETALDEHYDE

Abstract An RKR-based semiclassical inversion procedure for restricted rotation potentials is developed, tested, and applied to the torsional vibration of acetaldehyde. The method is based on a semiclassical quantization of the problem which in higher-order approximations is shown to yield eigenvalues of extremely high accuracy. Two-phase integral functions are extracted from the torsional eigenvalue spectrum, one of which depends on the potential well and the other on the potential barrier, and these two functions are then subjected to RKR manipulation. The model potential is reproduced within a few per cent in all regions of the potential and the height of the barrier of acetaldehyde, V 3 = 406.1 cm −1 , is in excellent agreement with the quantum mechanical results.

Semiclassical inversion of torsional vibration energy levels: Method, and application to acetaldehyde

An RKR-based semiclassical inversion procedure for restricted rotation potentials is developed, tested, and applied to the torsional vibration of acetaldehyde. The method is based on a semiclassical quantization of the problem which in higher-order approximations is shown to yield eigenvalues of extremely high accuracy. Two-phase integral functions are extracted from the torsional eigenvalue spectrum, one of which depends on the potential well and the other on the potential barrier, and these two functions are then subjected to RKR manipulation. The model potential is reproduced within a few per cent in all regions of the potential and the height of the barrier of acetaldehyde, V 3 = 406.1 cm −1, is in excellent agreement with the quantum mechanical results.

Semiclassical inversion of a symmetric double well potential

An RKR based inversion procedure for a symmetric double well potential is developed and tested for a model system giving rise to four doubled energy levels below the barrier maximum. Two phase integral functions are extracted from the eigenvalue spectrum, one of which depends on the potential well and the other on the potential barrier and these two functions are then subjected to conventional RKR manipulation. The model potential is reproduced within a few per cent over most of the energy range and the height of the barrier is determined within one per cent.

Determination of theHe4-He4Repulsive Potential up to 0.14 eV by Inversion of High-Resolution Total—Cross-Section Measurements

High-resolution measurements of the ${\mathrm{He}}^{4}$-${\mathrm{He}}^{4}$ total scattering cross section are presented for reduced collision energies between 0.2 and 200 meV. Clear evidence for the Ramsauer-Townsend effect is observed and thirteen backward glory extrema are resolved. From these extrema we derive the energy dependence of the $s$ phase shift. Applying the semiclassical inversion method proposed by Miller we then compute the repulsive potential up to 0.14 eV.

A semiclassical inversion procedure for the internuclear distance dependence of diatomic properties

A semiclassical procedure for obtaining the internuclear distance dependence of diatomic properties (e.g., dipole moment) from measured expectation values of the property in specific vibrational—rotational states is outlined. It represents generalization of the Rydberg-Klein-Rees procedure for obtaining the internuclear potential energy, and thus is valid up to dissociation.